C.5 Other formulas

To calculate \(z\)-scores (Sect. 20.4), use
\[ z = \frac{\text{value of variable} - \text{mean of the distribution of the variable}}{\text{standard deviation of the distribution of the variable}}. \] \(t\)-scores are like \(z\)-scores. When the 'variable' is a sample estimate (such as \(\bar{x}\)), the 'standard deviation of the distribution' is a standard error (such as \(\text{s.e.}(\bar{x})\)).
The unstandardising formula (Sect. 20.8) is \(x = \mu + (z\times \sigma)\).
The interquartile range (IQR) is \(Q_3 - Q_1\), where \(Q_1\) and \(Q_3\) are the first and third quartiles respectively (or, equivalently, the \(25\)th and \(75\)th percentiles).
The smallest expected value (for assessing statistical validity when forming CIs and conducting hypothesis tests with proportions or ORs) is
\[ \frac{(\text{Smallest row total})\times(\text{Smallest column total})}{\text{Overall total}}. \]
The regression equation in the sample is \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the sample intercept and \(b_1\) is the sample slope.